Average Error: 23.6 → 6.0
Time: 2.2m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + i\right) + \left(i + \alpha\right)}}{\frac{\left(i + i\right) + \left(2.0 + \left(\beta + \alpha\right)\right)}{\frac{\beta + \alpha}{1}}}}\right)}{2.0} \le 5.551115123125782 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + i\right) + \left(i + \alpha\right)}}{\frac{\left(i + i\right) + \left(2.0 + \left(\beta + \alpha\right)\right)}{\frac{\beta + \alpha}{1}}}}\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (log (exp (+ 1.0 (/ (/ (- beta alpha) (+ (+ beta i) (+ i alpha))) (/ (+ (+ i i) (+ 2.0 (+ beta alpha))) (/ (+ beta alpha) 1)))))) 2.0) < 5.551115123125782e-17

    1. Initial program 62.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 28.9

      \[\leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Applied simplify28.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}}\]

    if 5.551115123125782e-17 < (/ (log (exp (+ 1.0 (/ (/ (- beta alpha) (+ (+ beta i) (+ i alpha))) (/ (+ (+ i i) (+ 2.0 (+ beta alpha))) (/ (+ beta alpha) 1)))))) 2.0)

    1. Initial program 14.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity14.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    4. Applied times-frac0.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.7

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right) \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    7. Applied associate-*l*0.7

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-log-exp0.7

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}\right)}}{2.0}\]

    10. Applied simplify0.5

      \[\leadsto \frac{\log \color{blue}{\left(e^{1.0 + \frac{\frac{\beta - \alpha}{\left(\beta + i\right) + \left(i + \alpha\right)}}{\frac{\left(i + i\right) + \left(2.0 + \left(\beta + \alpha\right)\right)}{\frac{\beta + \alpha}{1}}}}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))